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Lie Algebras Joachim Wehler Lie Algebras DRAFT, Release 1.23 January 16, 2019 2 I have prepared these notes for the students of my lectures and my seminar. The lectures and the seminar took place during the summer semesters 2017 and 2018 (repetition) and the winter semester 2018/19 at the mathematical department of LMU (Ludwig-Maximilians-Universitat)¨ at Munich. I thank the participants of the seminar for making some simplifications and corrections. Compared to the oral lecture in class these written notes contain some additional material. Please report any errors or typos to [email protected] Release notes: • Release 1.23: Overview in Part II added. • Release 1.22: Chapter 8: Added Lemma 8.17. • Release 1.21: Chapter 8: Minor revisions. • Release 1.20: Chapter 8: Minor revisions. • Release 1.19: Chapter 8: Added proof of Theorem 8.8, minor revisions. • Release 1.18: Chapter 7: Proposition 7.6 added. • Release 1.17: Chapter 8: Minor revisions. • Release 1.16: Minor revisions. • Release 1.15: Update Chapter 6. • Release 1.14: Update Chapter 5, minor additions. • Release 1.13: Update Chapter 5, Section 5.2. • Release 1.12: Update Chapter 5, minor corrections and additions. • Release 1.11: Figure 8.1 corrected • Release 1.10: Added Chapter 8 • Release 1.9: Added Chapter 7 • Release 1.8: Added Chapter 6 • Release 1.7: Added Chapter 5, minor corrections. • Release 1.6: Added Chapter 4. • Release 1.5: Update References. • Release 1.4: Added Chapter 3. • Release 1.3: Added Chapter 2. • Release 1.2: Update Chapter 1. • Release 1.1: General revision Chapter 1. Contents Part I General Lie Algebra Theory 1 Matrix functions ............................................... 3 1.1 Power series of matrices . .3 1.2 Jordan decomposition . 12 1.3 Exponential of matrices . 24 2 Fundamentals of Lie algebra theory .............................. 35 2.1 Definitions and first examples . 35 2.2 Lie algebras of the classical groups . 41 2.3 Topology of the classical groups . 49 3 Nilpotent Lie algebras and solvable Lie algebras ................... 65 3.1 Engel’s theorem for nilpotent Lie algebras . 65 3.2 Lie’s theorem for solvable Lie algebras . 75 3.3 Semidirect product of Lie algebras . 82 4 Killing form and semisimple Lie algebras ......................... 91 4.1 Traces of endomorphisms . 91 4.2 Fundamentals of semisimple Lie algebras . 96 4.3 Weyl’s theorem on complete reducibility of representations . 106 Part II Complex Semisimple Lie Algebras 5 Cartan decomposition ..........................................127 5.1 Toral subalgebra . 127 5.2 sl(2;C)-modules . 130 5.3 Cartan subalgebra and root space decomposition . 140 6 Root systems ..................................................149 6.1 Abstract root system . 149 6.2 Action of the Weyl group . 159 v vi Contents 6.3 Coxeter graph and Dynkin diagram. 166 7 Classification ..................................................183 7.1 The root system of a complex semisimple Lie algebra . 183 7.2 Root systems of the A;B;C;D-series . 198 7.3 Outlook . 217 8 Irreducible Representations .....................................219 8.1 The universal enveloping algebra . 219 8.2 General weight theory . 236 8.3 Finite-dimensional irreducible representations . 246 References .........................................................257 Index .............................................................261 Part I General Lie Algebra Theory Chapter 1 Matrix functions The paradigm of Lie algebras is the vector space of matrices with the commutator of two matrices as Lie bracket. These concrete examples even cover all abstract finite dimensional Lie algebra which are the focus of these notes. Nevertheless it is useful to consider Lie algebras from an abstract viewpoint as a separate algebraic structure like groups or rings. If not stated otherwise, we denote by K either the field R of real numbers or the field C of complex numbers. Both fields have characteristic 0. The relevant differ- ence is the fact that C is algebraically closed, i.e. each polynomial with coefficients from K of degree n has exactly n complex roots. This result allows to transform matrices over C to certain standard forms by transformations which make use of the eigenvalues. 1.1 Power series of matrices At highschool every pupil learns the equation exp(x) · exp(y) = exp(x + y) This formula holds for all real numbers x,y. Then exponentiation defines a map ∗ exp : R ! R : Students who attended a class on complex analysis will remember that expo- nentiation is defined also for complex numbers. Hence exponentiation extends to a map ∗ exp : C ! C : This map satisfies the same functional equation. The reason for the seamless transition from the real to the complex field is the fact that the exponential map is defined by a power series 3 4 1 Matrix functions ¥ 1 exp(z) = ∑ · zn n=0 n! which converges not only for all real numbers but for all complex numbers too. In order to generalize further the exponential map we try to exponentiate argu- ments which are strictly upper triangular matrices. Example 1.1 (Exponential of strictly upper triangular matrices). We denote by 800 ∗ ∗1 9 < = n(3;K) := @0 0 ∗A 2 M(3 × 3;K) = (ai j)i j 2 M(n × n;K) : ai j = 0 if i ≥ j : 0 0 0 ; the K-algebra of strictly upper triangular matrices. Matrices from n(3;K) can be added and multiplied by each other, and they can be multiplied by scalars from the field K. In particular, we consider the matrices 00 p 01 00 0 01 P = @0 0 0A;Q = @0 0 qA 2 n(3;K): 0 0 0 0 0 0 We compute 0 1 0 1 ¥ 1 1 p 0 ¥ 1 1 0 0 exp(P) = ·Pn = 1+P = @0 1 0A;exp(Q) = ·Qn = 1+Q = @0 1 qA: ∑ n! ∑ n! n=0 0 0 1 n=0 0 0 1 Note Pn = Qn = 0 for all n ≥ 2. Hence we do not need to care about questions of convergence. i) To test the functional equation we compute on one hand 01 p 01 01 0 01 01 p pq1 exp(P) · exp(Q) = @0 1 0A · @0 1 qA = @0 1 q A = 1 + P + Q + PQ: 0 0 1 0 0 1 0 0 1 ii) On the other hand we compute exp(P + Q): For the computation of matrix products it is often helpful to introduce the basis (Ei j)1≤i; j≤n of the K-vector space M(n × n;K) with the matrices Ei j 2 M(n × n;K) having component = 1 at position with index i j and component = 0 at all other positions. Then 1.1 Power series of matrices 5 Ei j · Ekm = dim: We have 00 0 pq1 2 (P + Q) = (pE12 + qE23) · (pE12 + qE23) = pqE13 = @0 0 0 A = PQ 0 0 0 and (P + Q)n = 0; n ≥ 3: We obtain 01 p (pq)=21 2 exp(P+Q) = 1+P+Q+1=2·(P+Q) = 1+P+Q+(1=2)PQ = @0 1 q A: 0 0 1 iii) Apparently, the functional equation is not satisfied: exp(P + Q) 6= exp(P) · exp(Q): Fortunately, the defect can be repaired by introducing as correction term the matrix 0 1 0 0 (pq)=2 pq C := 1=2 · [P;Q] = @0 0 0 A = E13 = (1=2)PQ: 2 0 0 0 Here [P;Q] := PQ − QP denotes the commutator of the matrices P and Q. Computing PQ = pE12 · qE23 = pqE13; QP = qE23 · E13 = 0 we obtain [P;Q] = pqE13 2 (P+Q+C) = (pE12 +qE23 +(pq=2)E13)·(pE12 +qE23 +(pq=2)E13) = pqE13 = 2C and (P + Q +C)n = 0; n ≥ 3: Hence exp(P+Q+1=2·[P;Q]) = exp(P+Q+C) = 1+(P+Q+C)+(1=2)(P+Q+C)2 00 p pq1 = 1 + P + Q + 2C = 1 + P + Q + PQ = @0 0 q A = exp(P) · exp(Q): 0 0 0 iv) More general one can show: The exponential map can be defined on the strict upper triangular matrices 6 1 Matrix functions exp : n(3;K) ! GL(3;K): It satisfies the functional equation in the generalized form: For A; B 2 M(n×n;K)) exp(A) · exp(B) = exp(A + B + (1=2)[A;B]) The goal of the present section is to extend the exponential map to all matrices. Therefore we need a concept of convergence for sequences and series of matrices. The basic ingredience is the norm of a matrix. n Definition 1.2 (Operator norm). We consider the K-vector space K with the Eu- clidean norm s n 2 n kxk := ∑ jxij f or x = (x1;:::;xn) 2 K : i=1 On the K-algebra M(n × n;K) we introduce the operator norm n kAk := supfkAxk : x 2 K and kxk ≤ 1g: Note that kAk < ¥ due to compactness of the unit ball n fx 2 K : kxk ≤ 1g: Intuitively, the operator norm of A is the volume to which the linear map determined n by A blows-up or blows-down the unit ball of K . The operator norm kAk does only depend on the endormorphism which is repre- sented by the matrix A. Hence, all matrices which represent a given endormorphism with repect to different bases have the same operator norm. The K-vector space M(n × n;K) of all matrices with components from K is an associative K-algebra with respect to the matrix product A · B 2 M(n × n;K) because (A · B) ·C = A · (B ·C) for matrices A;B;C 2 M(n × n;K). Proposition 1.3 (Normed associative matrix algebra). The matrix algebra (M(n × n;K);kk) is a normed associative algebra: 1.
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